Soil Thermal Conductivity: Effects

of Saturation and Dry Density

B.A. Fricke A. Misra, Ph.D. B.R. Beeker W.E. Stewart, Jr.

ABSTRACT The effect of soil dry density on thermal conductiVity

is studied by means of a mathematical model that was
Saturation and dry density are important parameters developed for predicting conductivity at low saturations.
governing soil thermal conductivity. An increare in either This model is based upon a microstructural approach and
saturation or dry density of a soil will result in an increase experimental data. Predictions generated by this model are
in its conductivity. Both of these parameters should be compared with experimental results for low saturation levels
accounted for in conductivity prediction methods. In this in sandy soils.
paper, empirical correlations are presented that show the
dependence of thermal conductivity upon saturation. Also 1 LITERATURE SURVEY
a theoretical model is presented that gives an explicit
dependence of conductivity upon dry density. In addition, Several soil thermal conductivity prediction methods
this paper presents a review of existing prediction methods exist in the literature. These include Van Rooyen and
as well as a description of a data base of soil thermal Winterkom (1957), Johansen (1975), De Vries (1952),
conductivity measurements. Gemant (1952), and Kersten (1949). These methods vary in
applicability and complexity. A brief survey of these
INTRODUCTION various methods is given below.
Van Rooyen and Winterkom's (1957) correlation,
A soil's thermal conductivity is significantly influenced based on data collected from sands and, gravels, is given as
by its saturation and dry density. Saturation describes the follows:
amount of moisture contained in a soil, while dry density
(1)
refers to the mass of soil particles per unit volume. An
~ncrease in either the saturation or dry d'ensity of a soil will
result in an increase in its thermal conductivity. Other where
factors that have a secondary effect upon soil thermal soil thermal conductivity;
k
conductivity include mineral composition, temperature',
Sr degree of saturation; and,
texture, and time (Kersten 1949; Penner et al. 1975;
A,B,s functions of dry density, mineral type, and
Salomone et al. 1984; Salomone and Kovacs 1984; Salo-
granulometry respectively.
I
mone and Marlow 1989; Brandon and Mitchell 1989;
Mitchell 1991; Becker et al. 1992). The purpose of this The Van Rooyen-Winterkom method is limited to
paper is to investigate the influences of saturation and dry unfrozen sands and gravels with saturation levels between
density. Both of these parameters should be accounted for 1.5% and 10%.
'in soil thermal conductivity prediction algorithms. Johansen's (1975) correlation, which is based on
This paper presents soil thennal conductivity cor- thermal conductivity data for dry and saturated states at the
relations that were developed from measured data available same dry density, has the following form:
in the literature. Due to the great impact that soil moisture
k = (kSAT - kD ) • ke + kD (2)
content has on thermal conductivity, these correlations focus
upon conductivity as a function of saturation. The cor- where
relations were developed for five soil types, namely,
soil thermal conductivity;
gravels, sands, silts, clays, and peats, in both the frozen
soil thermal conductivity in the saturated
and unfrozen states. These soil types correspond to those
and dry states, respectively; and
used in the Unified Soil Classification System (USCS).
a dimensionless function of soil satura-
More exact soil classification into subclasses requires
tion.
detailed knowledge of the soil's grain size distribution and
its Atterberg Limits, namely, its liquid limit and its plastic Johansen's method is suitable for calculating soil
limit. Since this detailed information is not generally thermal conductivity of both coarse- and fine-grained soils
available in the literature, the present paper considers only in the frozen and unfrozen states, However, it is limited to
the loosely defined USCS classification. saturations greater than 20 %.
Brian A. Fricke is a research assistant, Bryan R. Becker is an assistant professor, and William E. Stewart, Jr., is a professor in the
Mechanical and Aerospace Engineering Department, University of Missouri-Kansas City at Independence, MO. Anil Misra is an assistant
professor in the Civil Engineering Department.
158
 The correlation given hy De Vries (1952) assumes that (5b)
a = 0.078s 112 ,
soil is a two,.phase material composed of uniform ellipsoidal
particles dispersed in a fluid phase. The De Vries cor- 0.16 X 1O- 3 sw-ho , (5c)
h
relation is given as

k (3) z = [1:ar [~r (5d)

where
I,s
x
fluid and solid phases, respectively;
volume fraction; and
b 2 -- [_ a
1 a
r [~r (5e)

k soil thermal conductivity. In Equation 5, s is soil dry density, w is moisture

content, h is the apex water (water collected around the
The factor F is given by contact points), ho is water absorbed as a film around the
soil particles, k, is the thermal conductivity of the solids,

F = ~~ [I + [~~ -1 1gir ' i = a,b,c . (4)

and kw is the thermal conductivity of water. Gemant's
method gives reasonable results for unfrozen sandy solis
only.
In Equation 4, the g values, which sum to unity, were Kersten (1949) tested many soil types and based his
originally intended to be shape factors but are usually used correlations on the empirical data he collected. He produced
to fit empirical data. De Vries' method is applicable to un- equations for frozen and unfrozen silt-clay soils and sandy
frozen coarse soils with saturations between 10% and 20%. soils. Kersten's correlations for unfrozen and frozen silt-
Gemant's (1952) correlation is based upon an idealized clay soils are as follows:
geometrical model of soil particles with point contacts. as
Unfrozen:
depicted in Figure 1. Water is assumed to_ collect around
O.Ol~d (6a)
the contact points to form a thermal bridge with heat flow k = [0.9Iogw - 0.2)10 .
assumed to be vertically upward. Gemant's correlation is
Frozen:
given as follows:
k = 0.01(IO)o.022~d + 0.085(IO)o.008~dw. (6b)
1 [(I_a)la)4/3 arc tan [(ks-kw)lkw)1I2
k The correlations for sandy soils are as follows:
(Sa) Unfrozen:
O.OI~d (7a)
k = [0.7Iogw + 0.4)10 .
Frozen:
k = 0.076(10)O.OI3~d + 0.032(10)o.OI46~d w. (7b)

In Equations 6 and 7, k is soil thermal conductivity

(Btu·in.1ft2·h·oF), w is moisture content, .and Yd is dry
density. Kersten's correlations give reasonable results only
for frozen soils with saturations up to 90 %.
Farouki (1986) has studied the applicability of these
methods and has suggested the conditions under which each
method should be used. It is clear that these methods are
applicable only for limited soil types and conditions, as
I-z
shown in Table 1. Hence, they do not offer a unified
I methodology for the estimation of soil thermal conductivity
applicable to a wide range of soil types and conditions.
I Therefore, these existing methods cannot be incorporated
into numerical heat transfer algorithms.
I In contrast, the correlations developed in this paper

~L
provide a unified methodology for evaluating soil thermal
conductivity. These correlations are applicable to soils in
five textural classes, namely, gravels, sands. silts, clays.
and peats, in both the frozen and unfrozen b"''''··~'''''i<'
their unified format, these new correlations
Figure 1 Idealized soil particle used in Gemant's cor-
incorporated into numerical heat transfer ~lg01;iIIm!i$:
relation (after Farouki, 1986).

159
 TABLE 1
Applicability of Prediction Metbods"
.

State Texture Saturation Method

Unfrozen Coarse Grained 0.Q15 - 0.100 Van Rooyen and Winterkorn (except for low-quartz crushed rock)
0.100 - 0.200 De Vries
0.200 - 1.000 Johansen
0.000 - 1.000 Gemant (sandy silt-clay)
saturated Johansen, De Vries, Gemant
Fine Grained 0.000 - 0.100 Johansen (underpredicts by 15%)
0.100 - 0.200 Johansen (underpredicts by 5%)
0.200 - 1.000 Johansen
saturated Johansen, De Vries, Gemant
Frozen Coarse Grained 0.100 - 1.000 Johansen
saturated Johansen, De Vries
Fine Grained 0.000 - 0.900 Kersten
0.100-1.000 Johansen
saturated Johansen. De Vries

'Data from Farouki (1986).

DEVELOPMENT OF DATA BASE classified into five general types-gravel, sand, silt, clay,
and peat. A brief description of each of the five soil
In order to develop empirical correlations for soil samples that constitute the data base is given below.
thermal conductivity, a data base was created from mea-
sured data available in the literature. The measured soil Gravel
thermal conductivity data reported in the literature were
Most of the measured data on gravels are from Kersten
obtained by perfonrung either a steady-state or a transient
(1949). These data include Chena River gravel, which is
test.
mainly composed of quartz and igneous rock with sizes
In the steady-state method, a temperature gradient is
ranging from 0.10 to 0.75 in. (2.5 to 19 mm).
applied to a soil sample until constant heat flow is obtained.
Knowledge of the temperature gradient across the soil
Sand
sample allows for the calculation of its thermal conduc-
tivity. Steady-state testing is time consuming and, because The measured data on sand were collected from the
of this, the soil sample is susceptible to moisture diffusion. works of Kersten (1949), Salomone and Marlow (1989), De
The resulting loss of moisture will affect the heat flow and Vries (1952), Andersland and Anderson (1978), Nakshaban-
thus the thermal conductivity (Kersten 1949; Penner et a!. di and Kohnke (1965), and Sawada (1977).
1975; Farouki 1986). Of the data sources cited in this Kersten presented data on 12 sand samples, of which
paper, only Kersten made use of the steady-state test. five were natural sands .and seven were man-made. The five
The transient method involves inserting a thin, con- natural sands include Fairbanks sand, Lowell sand, North-
stant-flux heat probe into a soil sample. By knowing the way sand, Northway fine sand, and Dakota sandy loam.
heat flux and soil temperature history, the soil thermal The Fairbanks sand was a siliceous sand with 27.5 % of the
conductivity can be calculated. Due to the shorter time particles larger than 0.079 in. (2.0 mm) and 70% of the
requirement, moisture migration is decreased in the tran- particles between 0.020 and 0.079 in. (0.5 and 2.0 mm).
sient test as compared to the steady-state test. This usually The Lowell sand was also siliceous, with particles between
results in a more accurate measurement of soil thermal 0.02 and 0.079 in. (0.5 and 2.0 mm). The two Northway
conductivity (penner et al. 1975; Salomone et al. 1984; sands are similar in composition. their main constituent
Salomone and Kovacs 1984; Salomone and Marlow 1989; . being feldspar with grain sizes ranging from 0.19 to 0.0030
Farouki 1986). in. (4.75 mm to 0.075 mm). No details are available on the
In the work described in this paper, thermal conduc- Dakota sandy loam.
tivity data at various dry densities, moisture contents, and Of the seven man-made sands, three were feldspar
temperatures were collected for each soil type. To obtain sands and four were quartz sands. The feldspar sands
reasonable results, many sources of data were consulted: consisted 0[90% sand-sized particles and 10% gravel-sized
Kersten (1949), Penner et al. (1975), Salomone and Marlow particles. The quartz sands included one sample with grain
(1989), De Vries (1952), Farouki (1986), Andersland and sizes larger than 0.020 in. (0.5 mm) and three samples with
Anderson (1978), Nakshabandi and Kohnke (1965), and grain sizes between 0.020 and 0.079 in. (0.5 mm to 2.0
Sawada (1977). Based upon texture, the soil data were mm).

160
 The sands tested by Salomone and Marlow (1989) were Moisture content, w, and saturation, S, are given as
classified according to the Unified Soil Classification follows:
w= (9a)
System (USCS). These sands included well-graded sands
(SW), poorly graded sands (SP), silty sands (SM), and
clayey sands (sq. However, no information was available (9b)
Vw
concerning their mineral constituents. S =-
where Vv
The remaining sands were fine-grained sands; however,
no information is available on their grain size distributions Mw = mass of water,
or mineral constituents. Vw = volume of water, and
VV = volume of void spaces.

Silt Combining Equations 8 and 9 yields the following expres-

sion for saturation, in which Pw is the density of water:
The measured data on silt are from Kersten (1949) and
Salomone and Marlow (1989). Kersten tested three silts: x 100%.
Northway silt loam, Fairbanks silt loam, and Fairbanks silty (10)
clay loam. All three silts were classified as low-plasticity
silts (ML) according to the USCS. Salomone and Marlow
presented data for several low-plasticity silts. Little infor-
mation is available on the mineral constituents of these silts. Thermal Conductivity vs. Saturation
As depicted in Figure 2, the thermal conductivity of a
Clay soil increases in three stages as the saturation level increas-
The measured data on clay are from Kersten (1949), es. At low saturations, moisture first coats the soil particles.
Salomone and Marlow (1989), and Penner et al. (1975).
Kersten tested two clays-Ramsey sandy loam and Healy
clay-both of which were classified as low-plasticity clays
(CL). The main mineral constituent of these clays is
kaolinite. Salomone and Marlow tested both high- and low- STAGE 1
plasticity clays; however, no infonnation was given concer-
ning the mineral composition of these clays. The clay
samples tested by Penner et al. were low-plasticity clays
containing quartz, illite, chlorite, and kaolinite.

Peat
The measured data on peat are from Kersten (1949) and
Salomone and Marlow (1989). Kersten tested Fairbanks
STAGE 2
peat, while Salomone and Marlow tested highly decomposed
woody peat.

EFFECTS OF SATURATION

Basic Definitions
An expression for saturation can be derived from the
basic definitions of dl Y density, solid density, and moisture
content. Dry density, Pd' and solid density, Ps' are defined
as follows:
STAGE 3
(8a)

(8b)
Ps =
Vs '
where
Figure 2 Saturation states of granular media:
Ms mass of solid soil particles, Stage I-moisture barely coats the particles;
Vs = volume of the solid particles, and Stage 2-moisture collects at particle contacts,-
VT total volume. Stage 3-moisture fills the void space.
The gaps between the soil particles are not filled rapidly amount of measured data for peaty soils, only a mean
and thus there is a slow increase in thennal conductivity. correlation is presented. Measured data collected for gravel
When the particles are fully coated with moisture, a further include saturations up to approximatel y 40 % and, thus, the
increase in the moisture content fills the voids between correlations for gravel are valid only to 40% saturation.
particles. This increases the heat flow between particles, An error analysis of these correlations is presented in
resulting in a rapid increase in thennal conductivity. the work by Becker et al. (1992). The difference, Z,
Finally, when all the voids are filled, further increasing the between the mean correlation and the measured data was
moisture content no longer increases the heat flow, and the calculated at each data point. A normalized di.fference, z",
thermal conductivity does not appreciably increase. The was calculated as Z' = (Z -7:)I<Iz ' in which Z is the mean
model used to describe this behavior is as follows: of the calculated differences and <Iz is the standard deviation
S = Adsinh(A2k + A3) - sinh(A4)] (11) of those differences. The cumulative frequency of the
normalized difference, Z', was compared to a cumulative
where nonnal distribution fu~ction. This error analysis shows that
these correlations provide a good fit to the measured data.
S saturation,
k soil thermal conductivity (Btu· in.! EFFECTS OF DRY DENSITY
ft 2·h·oF), and
coefficients that depend upon soil At any given saturation level, the soil thermal conduc-
type. tivity exhibits considerable variation, as shown in Figures
3 through 7. This variation is due, in part, to differences in
The values of Al through A4 for each of the five soil dry density. The effect of dry density upon soil thermal
types in both the frozen and unfrozen states are given in conductivity was studied by means of a mathematical model
Table 2. At a saturation of zero, Equation II reduces to the that was developed for a particulate system composed of
following: random arrays of identical spheres in an almost dry state,
(12) as depicted in stages I and 2 of Figure 2. The heat transfer
in this simple system can be idealized to occur in the
Equation 12 shows that the coefficient A4 is related to neighborhood of the interparticle contacts. Based upon this
the thermal conductivity of dry soil, 1<0. model, under low confining stress, the expression for
Figures 3 through 7 present the measured soil thermal thermal conductivity as a function of dry density was found
conductivity versus saturation data for the five soil types in to be (Misra et al. 1992)
both the frozen and unfrozen states. The empirical cor-
relations, based upon Equation 11, are also plotted in
Figures 3 through 7. Three curves have been given for each
soil type (except peat). The upper curve represents the
upper limit of the measured data, the middle curve is the
mean of the measured data, and the lower curve represents Equation 13 shows that thermal conductivity varies
the lower limit of the measured data. Due to the small linearly with dry density. The measured data exhibit a

TABLE 2
Correlation Coefficients
Frozen Al A2 A3 A4
Soil
Type Unfrozen Low Mid High Low Mid High Low Mid High Low Mid High

Clay Frozen 23.5 14.5 14.0 0.25 0.25 0.25 -2.0 -2.5 -3.0 -1.75 -2.0 -2.0
Unfrozen 33.5 27.0 14.0 0.29 0.265 0.32 -1.6 -1.5 -3.0 -1.31 -0.97 -1.72
Gravel Frozen 25.4 11.0 11.3 0.29 0.35 0.3 -2.1 -3.0 -2.8 -1.23 -1.6 ·0.85
Unfrozen 16.5 6.5 8.3 0.32 0.38 0.2 -1.9 -3.0 -1.8 -1.1 -1.48 -0.8
Peat Frozen 12.0 0.4 -2.6 -2.52
Unfrozen 28.0 0.865 -1.9 -1.4675
Sand Frozen 26.0 10.0 15.0 0.265 0.24 0.17 -1.0 -2.2 -1.8 -0.735 -1.625 -0.44
Unfrozen 6.4 6.8 6.8 0.8 0.4 0.5 -3.2 -2.9 -7.5 -2.0 -1.5 -2.0
Silt Frozen 38.0 19.5 18.5 0.24 0.27 0.2 -1.2 -1.8 -2.0 -0.96 -1.53 -1.8
Unfrozen 28.0 17.0 22.0 0.4 0.4 0.25 -1.0 -2.6 -2.2 -0.6 -1.6 ·0.95

162
 25.0

...... . ..
'

20.0
.
~
.
)

15.0
..
10.0 ..
I ..

5.0

0.0 20.0 ~o.o 60.0 80.0 100.0 0.0 20.0 '10.0 60.0 80.0

5(%)
5(%)
(a)
(a)

20.0 25.0

i2 ,
15.0
o
. 20.0
..., ..
~
........... 15.0
..
.5, ..
10.0
.2
6
~
10.0
r
. {.
5.0
. 5.0

0.0 20.0 ~o.o 60.0 80.0 100.0 0.0 20.0 10.0 60.0 80.0 100.0

5(%) 5(%)
(b) (b)

Figure 3 Thermal COluiuctivity vs. saturationfor gravel: Figure 4 Thermal conductivity vs. saturation for sand:
(a) frozen; (b) unfrozen. (a) frozen; (b) unfrozen.

similar behavior. Also, Equation 13 accounts for the CONCLUSIONS

presence of moisture at low levels. This behavior is similar
to the experimental results reported by Brandon and This paper has focused upon the influence that soil
Mitchell (1989) for sands with low moisture content. In saturation and dry density have on thermal conductivity. To
Figure 8, a comparison is presented between the predictions this end, a family of empirical correlations was presented
and the measurements for thermal conductivity VS. dry that relates thermal conductivity to saturation. The effects
density for quartz sands at three levels of moisture content: of dry density were investigated by means of a microstruc-
w = 0.0%, 0.5%, and 1.0%. In these predictions, the tural, theoretical model. Taken together, the empirical
thermal conductivity of a solid particle is taken to be 58.25 correlations and the theoretical model correctly represent
Btu·in.1ft2·h·oF (8.39 W/m'oC) and that of air is taken to the influence of saturation and dry density upon soil thermal
be 0.18 Btu·in./fr·h·oF (0.026 W/m·°C). The parameter, conductivity.
b, is set to -3.1 and c is set to 10.5. As shown in Figure 8, This paper also presented a review of existing predic-
a good agreement exists between the predictions and the tion methods as well as a description of a data base of
measured data. thennal conductivity measurements.

163
 20.0 25.0

20.0
,
15.0

I
-. ~... .. .'
15.0
, ... :
:,' ~
10.0 .. , ~
.,
" 10.0

5.0
5.0

"
0.0 80.0 100.0
0.0 20.0 10.0 60.0 80.0 100.0 0.0 20.0 10.0 60.0

S(%) S(%)
(a) (a)

20.0 20.0

,
~

G:
0, 15.0
>I-<
0, 15.0 •
.c>-<, ,, .c>-<
i:'- ",
'"i..
~

'-
"
10.0 ", ,
..
.
.5, 10.0 " f .: .... .5, , , I
I. , • '.
.2 ' " .2 , , •
, ~
e..I<: po ,
5.0 ,
, ,
, .
•
)Z
5.0
,,
-, , ,
~ "
"
I

60.0 80.0 100.0 0.0 20.0 10.0 60.0 80.0 100.0

0.0 20.0 10.0

S(%) S(%)
(b) (b)
Figure 5 Thermal conductivity vs. saturationfor silt: (a) Figure 6 Thermal conductivity vs. saturation for clay:
frozen; (b) unfrozen. (a) frozen; (b) unfrozen.

ingen 52(1):1-73. Translated by Building Research

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164
 .~O-- w = 1.0%
10.0
~

f-t.
0 , G;
.... 01-
~ , 15.0
~
......
i w;= 0.5%
.,. ~ D B
<=; D
10,0 k" 5.0 D
.2
(0
6
-" C
~

..:.: D
EI3 DC!
w;= 0.0%
C
5.0 D
C C

0.0
0.0 90.0 100.0 110.0 120.0
0.0 20.0 10.0 60.0 80.0 100.0
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S(%)
(a) Figure 8 Thennal conductivity vs. dry density for quartz
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~

f-t.
0 , tension and other physical properties. Agricultural
.... 15.0
~ , Meteorology 2:271-279.
'1..
"-< Penner, E., G.H. Johnston, and L.E. Goodrich. 1975.
......
.S, Thermal conductivity laboratory studies of some
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.2
(0
~ Journal 12(3):271-288. August.
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165